Abstract In deterministic global optimization, techniques for linear relaxation of a non-convex program are used in the lower bound calculation phase. To achieve this phase, most deterministic global optimization codes use reformulation-linearization techniques. However, there exist also two interval-based polyhedral relaxation techniques which produce reliable bounds without adding new auxiliary variables, and which can take into account mathematical operations and most transcendental functions: (i) the affine relaxation technique, used in the IBBA code, based on affine forms and affine arithmetic, and (ii) the extremal Taylor technique, used in the Ibex-Opt code, which is based on a specific interval-based Taylor form. In this paper, we describe how these two interval-based linear relaxation techniques can be hybridized. These two approaches appear to be complementary, and such a hybrid method performs well on a representative sample of constrained global optimization instances.

Proving that the state of a controlled nonlinear system always stays inside a time moving bubble (or capture tube) amounts to proving the inconsistency of a set of nonlinear inequalities in the time-state space. In practice however, even with a good intuition, it is difficult for a human to find such a capture tube except for simple examples. In 2014, Jaulin et al. established properties that support a new interval approach for validating a quasi capture tube, i.e. a candidate tube (with a simple form) from which the mobile system can escape, but into which it enters again before a given time. A quasi capture tube is easy to find in practice for a controlled system. Merging the trajectories originated from the candidate tube yields the smallest capture tube enclosing it. This paper proposes an interval constraint programming solver dedicated to the quasi capture tube validation. The problem is viewed as a differential CSP where the functional variables correspond to the state variables of the system and the constraints define system trajectories that escape from the candidate tube "for ever". The solver performs a branch and contract procedure for computing the trajectories that escape from the candidate tube. If no solution is found, the quasi capture tube is validated and, as a side effect, a corrected smallest capture tube enclosing the quasi one is computed. The approach is experimentally validated on several examples having 2 to 5 degrees of freedom.

This paper presents a new interval-based operator for continuous constrained global optimization. It is built upon a new filtering operator, named TEC, which constructs a bounded subtree using a Branch and Contract process and returns the parallel-to-axes hull of the leaf domains/boxes. Two extensions of TEC use the information contained in the leaf boxes of the TEC subtree to improve the two bounds of the objective function value: (i) for the lower bounding, a polyhedral hull of the (projected) leaf boxes replaces the parallel-to-axes hull, (ii) for the upper bounding, a good feasible point is searched for inside a leaf box of the TEC subtree, following a look-ahead principle. The algorithm proposed is an auto-adaptive version of TEC that plays with both extensions according to the outcomes of the upper and lower bounding phases. Experimental results show a significant speed-up on several state-of-the-art instances.

The parameter estimation problem is awidespread and challenging problem in engineering sciencesconsisting in computing the parameters of a parametricmodel that fit observed data. Calibration or geolocation canbe viewed as specific parameter estimation problems. In thispaper we address the problem of finding all the instances ofa parametric model that can explain at least q observationswithin a given tolerance. The computer vision communityhas proposed the RANSAC algorithm to deal with outliersin the observed data. This randomized algorithm is efficientbut non-deterministic and therefore incomplete. Jaulin etal. proposes a complete and combinatorial algorithm thatexhaustively traverses the whole space of parameter vectors toextract the valid model instances. This algorithm is based oninterval constraint programming methods and on a so calledq-intersection operator, a relaxed intersection operator thatassumes that at least q observed data are inliers. This paperproposes several improvements to Jaulin et al.'s algorithm.Most of them are generic and some others are dedicated tothe shape detection problem used to validate our approach.Compared to Jaulin et al.'s algorithm, our algorithm canguarantee a number of fitted observations in the producedmodel instances. Also, first experiments in plane and circlerecognition highlight speedups of two orders of magnitude.

Singleton-based consistencies have been shown to dramatically improve the performance of constraint solvers on some difficult instances. However, they are in general too expensive to be applied exhaustively during the whole search. In this paper, we focus on partition-one-AC, a singleton-based consistency which, as opposed to singleton arc consistency, is able to prune values on all variables when it performs singleton tests on one of them. We propose adaptive variants of partition-one-AC that do not necessarily run until having proved the fixpoint. The pruning can be weaker than the full version but the computational effort can be significantly reduced. Our experiments show that adaptive Partition-one-AC can obtain significant speed-ups over arc consistency and over the full version of partition-one-AC.

Given a set of axis-parallel n-dimensional boxes, the q-intersection is defined as the smallest box encompassing all the points that belong to at least q boxes. Computing the q-intersection is a combinatorial problem that allows us to handle robust parameter estimation with a numerical constraint programming approach. The q-intersection can be viewed as a filtering operator for soft constraints that model measurements subject to outliers. This paper highlights the equivalence of this operator with the search of q-cliques in a graph whose boxicity is bounded by the number of variables in the constraint network. We present a computational study of the q-intersection. We also propose a fast heuristic and a sophisticated exact q-intersection algorithm. First experiments show that our exact algorithm outperforms the existing one while our heuristic performs an efficient filtering on hard problems.

An operator called CID and an efficient variant 3BCID wereproposed in 2007. For numerical CSPs handled by interval methods, these operators compute a partial consistency equivalent to Partition-1-AC for discrete CSPs. The two main parameters of CID are the number of times the main CID procedure is called and the maximum number ofsub-intervals treated by the procedure. The 3BCID operator is state-of-the-art in numerical CSP solving, but not in constrained global optimization. This paper proposes an adaptive variant of 3BCID. The number of variables handled is auto-adapted during the search, the other parameters are fixed and robust to modifications. On a representative sample of instances, ACID appears to be the best approach in solving and optimization, and has been added to the default strategies of the Ibex interval solver.

Researchers from interval analysis and constraint (logic) programming communities have studied intervals for their ability to manage infinite solution sets of numerical constraint systems. In particular, inner regions represent subsets of the search space in which all points are solutions. Our main contribution is the use of recent and new inner region extraction algorithms in the upper bounding phase of constrained global optimization. Convexification is a major key for efficiently lower bounding the objective function. We have adapted the convex interval taylorization proposed by Lin and Stadherr for producing a reliable outer and inner polyhedral approximation of the solution set and alinearization of the objective function. Other original ingredients are part of our optimizer, including an efficient interval constraint propagation algorithm exploiting monotonicity of functions. We end up with a new framework for reliable continuous constrained global optimization. Our interval B&B is implemented in the interval-based explorer Ibex and extends this free C++ library. Our strategy outperforms the best reliable global optimizers.

We propose in this paper a new interval constraint propagation algorithm, called MOnotonic Hull Consistency (Mohc), that exploits monotonicity of functions. The propagation is standard, but the Mohc-Revise procedure, used to filter/contract the variable domains w.r.t. an individual constraint, uses monotonic versions of the classical HC4-Revise and BoxNarrow procedures. Mohc-Revise appears to be the first adaptive revise procedure ever proposed in (interval) constraint programming. Also, when a function is monotonic w.r.t. every variable, Mohc-Revise is proven to compute the optimal/sharpest box enclosing all the solutions of the corresponding constraint (hull consistency). Very promising experimental results suggest that Mohc has the potential to become an alternative to the state-of-the-art HC4 and Box algorithms.

When handling 2D packing problems, numerous incomplete and complete algorithms maintain a so-called bottom-left (BL) property: every rectangle placed in a container is propped up bottom and left. While it is easy to make a rectangle BL when it is is added in a container, it is more expensive to maintain all the placed pieces BL when a rectangle is removed. This prevents researchers from designing incremental moves for metaheuristics or efficient complete optimization algorithms. This paper investigates the possibility of violating the BL property. Instead, we propose to maintain only the set of "maximal holes", which allows incremental additions and removals of rectangles. To validate our alternative approach, we have designed an incremental move, maintaining maximal holes, for the strip-packing problem, a variant of the famous 2D bin-packing. We have also implemented a generic metaheuristic using this move and standard greedy heuristics. Experimental results show that the approach is competitive with the best known incomplete algorithms, especially the other metaheuristics (able to escape from local minima).

We present a new approach to 3D scene modeling based on geometric constraints. Contrary to the existing methods, we can quickly obtain 3D scene models that respect the given constraints exactly. Our system can describe a large variety of linear and nonlinear constraints in a flexible way. To deal with the constraints, we decided to exploit the properties of the GPDOF algorithm developed in the Constraint Programming community (Trombettoni, 1998). The approach is based on a dictionary of so-called r-methods, based on theorems of geometry, which can solve a subset of geometric constraints in a very efficient way. GPDOF is used to find, in polynomial-time, a reduced parameterization of a scene, and to decompose the equation system, induced by constraints, into a sequence of r-methods. We have validated our approach in reconstructing, from images, 3D models of buildings based on linear and quadratic geometric constraints.